Abstract: My plan is to give a general survey of exchangeable and related random arrays, some of which arise naturally in the study of random networks. For three of the basic probabilistic symmetries, we can derive general representations that give a fairly complete understanding of independence and other structural properties. Though some of those results are extremely tricky to prove, I shall try to keep my talk on a rather elementary level that avoids the technical subtleties of the subject. (This talk was prepared for a workshop on random networks in New York next week, with participants from famous universities around the world.)